| L(s) = 1 | − 10.4·2-s + 9·3-s + 77.9·4-s − 94.3·6-s + 152.·7-s − 481.·8-s + 81·9-s + 121·11-s + 701.·12-s + 260.·13-s − 1.60e3·14-s + 2.55e3·16-s − 586.·17-s − 849.·18-s − 614.·19-s + 1.37e3·21-s − 1.26e3·22-s + 1.37e3·23-s − 4.33e3·24-s − 2.72e3·26-s + 729·27-s + 1.19e4·28-s − 4.06e3·29-s − 2.68e3·31-s − 1.13e4·32-s + 1.08e3·33-s + 6.15e3·34-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.43·4-s − 1.07·6-s + 1.17·7-s − 2.66·8-s + 0.333·9-s + 0.301·11-s + 1.40·12-s + 0.426·13-s − 2.18·14-s + 2.49·16-s − 0.492·17-s − 0.617·18-s − 0.390·19-s + 0.681·21-s − 0.558·22-s + 0.542·23-s − 1.53·24-s − 0.791·26-s + 0.192·27-s + 2.87·28-s − 0.896·29-s − 0.502·31-s − 1.96·32-s + 0.174·33-s + 0.912·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 10.4T + 32T^{2} \) |
| 7 | \( 1 - 152.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 260.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 586.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 614.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.30e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.54e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.73e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.23e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.970779690624769721229100637775, −8.370260135548802570355111721126, −7.65671564902208825555995612861, −6.96600054434470613955627628373, −5.89364318032810331877057677762, −4.46308109862332177505595571408, −3.06080289600461421282673557320, −1.87827805639411217032559007564, −1.36082101776373264380946692242, 0,
1.36082101776373264380946692242, 1.87827805639411217032559007564, 3.06080289600461421282673557320, 4.46308109862332177505595571408, 5.89364318032810331877057677762, 6.96600054434470613955627628373, 7.65671564902208825555995612861, 8.370260135548802570355111721126, 8.970779690624769721229100637775
